Related papers: From Sturm-Liouville problems to fractional and an…
The article is dedicated towards the study of fractional order non-linear differential systems with non-instantaneous impulses involving Riemann-Liouville derivatives with fixed lower limit and appropriate integral type initial conditions…
Very recently, some authors have studied new types of fractional derivatives whose kernels are nonsingular. In this article, we study Sturm-Liouville Equations ($SLEs$) in the frame of fractional operators with Mittag-Leffler kernels. We…
The anomalous (i.e. non-Gaussian) dynamics of particles subject to a deterministic acceleration and a series of 'random kicks' is studied. Based on an extension of the concept of continuous time random walks to position-velocity space, a…
The Liouville and first Bogoliubov hierarchy equations with derivatives of noninteger order are derived. The fractional Liouville equation is obtained from the conservation of probability to find a system in a fractional volume element.…
This review article aims to stress and reunite some of the analytic formalism of the anomalous diffusive processes that have succeeded in their description. Also, it has the objective to discuss which of the new directions they have taken…
We study the generalized Langevin equation approach to anomalous diffusion for a harmonic oscillator and a free particle driven by different forms of internal noises, such as power-law-correlated and distributed-order noises that fulfil…
Levy flights and subdiffusive processes and their properties are discussed. We derive the space- and time-fractional transport equations, and consider their solutions in external potentials. An extensive list of references is included.
We consider a half-order time-fractional diffusion equation in an arbitrary dimension and investigate inverse problems of determining the source term or the diffusion coefficient from spatial data at an arbitrarily fixed time under some…
In this chapter, we mainly review theoretical results on inverse source problems for diffusion equations with the Caputo time-fractional derivatives of order $\alpha\in(0,1)$. Our survey covers the following types of inverse problems: 1.…
A numerical study of the role of anomalous diffusion in front propagation in reaction-diffusion systems is presented. Three models of anomalous diffusion are considered: fractional diffusion, tempered fractional diffusion, and a model that…
Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…
The problem of anomalous diffusion in momentum (velocity) space is considered based on the master equation and the appropriate probability transition function (PTF). The approach recently developed by the author for coordinate space, is…
The Mellin transform is usually applied in probability theory to the product of independent random variables. In recent times the machinery of the Mellin transform has been adopted to describe the L\'evy stable distributions, and more…
We survey methods and results of fractional differential equations in which an unknown function is under the operation of integration and/or differentiation of fractional order. As an illustrative example, we review results on fractional…
In this paper continuous time random walk models approximating fractional space-time diffusion processes are studied. Stochastic processes associated with the considered equations represent time-changed processes, where the time-change…
In this paper we analyze a coupling between the very large jumps in physical and operational times as applied to anomalous diffusion. The approach is based on subordination of a skewed Levy-stable process by its inverse to get two types of…
This is a review of statistical inference methodology for stochastic differential equations driven by fractional Brownian motion, otherwise called fractional diffusions. The first section reviews the theory needed to rigorously define them.…
A class of optimal control problems governed by linear fractional diffusion equation with control constraint is considered. We first establish some results on the existence of strong solution to the state equation and the existence of…
We prove that any given function can be smoothly approximated by functions lying in the kernel of a linear operator involving at least one fractional component. The setting in which we work is very general, since it takes into account…
There has been considerable recent study in "sub-diffusion" models that replace the standard parabolic equation model by a one with a fractional derivative in the time variable. There are many ways to look at this newer approach and one…